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FR68,6\text{FR}^{8,6}_{6}

Fusion Rules

1234567821654387364125784513627854263187635214877877882+5+6+7+81+3+4+7+88788771+3+4+7+82+5+6+7+8\begin{array}{|llllllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} \\ \mathbf{2} & \mathbf{1} & \mathbf{6} & \mathbf{5} & \mathbf{4} & \mathbf{3} & \mathbf{8} & \mathbf{7} \\ \mathbf{3} & \mathbf{6} & \mathbf{4} & \mathbf{1} & \mathbf{2} & \mathbf{5} & \mathbf{7} & \mathbf{8} \\ \mathbf{4} & \mathbf{5} & \mathbf{1} & \mathbf{3} & \mathbf{6} & \mathbf{2} & \mathbf{7} & \mathbf{8} \\ \mathbf{5} & \mathbf{4} & \mathbf{2} & \mathbf{6} & \mathbf{3} & \mathbf{1} & \mathbf{8} & \mathbf{7} \\ \mathbf{6} & \mathbf{3} & \mathbf{5} & \mathbf{2} & \mathbf{1} & \mathbf{4} & \mathbf{8} & \mathbf{7} \\ \mathbf{7} & \mathbf{8} & \mathbf{7} & \mathbf{7} & \mathbf{8} & \mathbf{8} & \mathbf{2}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{1}+\mathbf{3}+\mathbf{4}+\mathbf{7}+\mathbf{8} \\ \mathbf{8} & \mathbf{7} & \mathbf{8} & \mathbf{8} & \mathbf{7} & \mathbf{7} & \mathbf{1}+\mathbf{3}+\mathbf{4}+\mathbf{7}+\mathbf{8} & \mathbf{2}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} \\ \hline \end{array}

The fusion rules are invariant under the group generated by the following permutations:

{(7 8),(3 4)(5 6)}\{(\mathbf{7} \ \mathbf{8}), (\mathbf{3} \ \mathbf{4}) (\mathbf{5} \ \mathbf{6})\}

The following elements form non-trivial sub fusion rings

Elements SubRing
{1,2}\{\mathbf{1},\mathbf{2}\} Z2: FR12,0\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}
{1,3,4}\{\mathbf{1},\mathbf{3},\mathbf{4}\} Z3: FR13,2\mathbb{Z}_3:\ \text{FR}^{3,2}_{1}
{1,2,3,4,5,6}\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4},\mathbf{5},\mathbf{6}\} Z6: FR16,4\mathbb{Z}_6:\ \text{FR}^{6,4}_{1}

Frobenius-Perron Dimensions

Particle Numeric Symbolic
1\mathbf{1} 1.1. 11
2\mathbf{2} 1.1. 11
3\mathbf{3} 1.1. 11
4\mathbf{4} 1.1. 11
5\mathbf{5} 1.1. 11
6\mathbf{6} 1.1. 11
7\mathbf{7} 3.3. 33
8\mathbf{8} 3.3. 33
DFP2\mathcal{D}_{FP}^2 24.24. 2424

Characters

The symbolic character table is the following

1243657811111133111111−1−11−111−1−1i3−i31−111−1−1−i3i31112(−1−i3)12(−1+i3)12(−1+i3)12(−1−i3)001112(−1+i3)12(−1−i3)12(−1−i3)12(−1+i3)001−112(−1+i3)12(−1−i3)12(1+i3)12(1−i3)001−112(−1−i3)12(−1+i3)12(1−i3)12(1+i3)00\begin{array}{|cccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{4} & \mathbf{3} & \mathbf{6} & \mathbf{5} & \mathbf{7} & \mathbf{8} \\ \hline 1 & 1 & 1 & 1 & 1 & 1 & 3 & 3 \\ 1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 \\ 1 & -1 & 1 & 1 & -1 & -1 & i \sqrt{3} & -i \sqrt{3} \\ 1 & -1 & 1 & 1 & -1 & -1 & -i \sqrt{3} & i \sqrt{3} \\ 1 & 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & 0 & 0 \\ 1 & 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & 0 & 0 \\ 1 & -1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(1+i \sqrt{3}\right) & \frac{1}{2} \left(1-i \sqrt{3}\right) & 0 & 0 \\ 1 & -1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(1-i \sqrt{3}\right) & \frac{1}{2} \left(1+i \sqrt{3}\right) & 0 & 0 \\ \hline \end{array}

The numeric character table is the following

124365781.0001.0001.0001.0001.0001.0003.0003.0001.0001.0001.0001.0001.0001.000−1.000−1.0001.000−1.0001.0001.000−1.000−1.0001.732i−1.732i1.000−1.0001.0001.000−1.000−1.000−1.732i1.732i1.0001.000−0.5000−0.8660i−0.5000+0.8660i−0.5000+0.8660i−0.5000−0.8660i001.0001.000−0.5000+0.8660i−0.5000−0.8660i−0.5000−0.8660i−0.5000+0.8660i001.000−1.000−0.5000+0.8660i−0.5000−0.8660i0.5000+0.8660i0.5000−0.8660i001.000−1.000−0.5000−0.8660i−0.5000+0.8660i0.5000−0.8660i0.5000+0.8660i00\begin{array}{|rrrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{4} & \mathbf{3} & \mathbf{6} & \mathbf{5} & \mathbf{7} & \mathbf{8} \\ \hline 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 3.000 & 3.000 \\ 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & -1.000 & -1.000 \\ 1.000 & -1.000 & 1.000 & 1.000 & -1.000 & -1.000 & 1.732 i & -1.732 i \\ 1.000 & -1.000 & 1.000 & 1.000 & -1.000 & -1.000 & -1.732 i & 1.732 i \\ 1.000 & 1.000 & -0.5000-0.8660 i & -0.5000+0.8660 i & -0.5000+0.8660 i & -0.5000-0.8660 i & 0 & 0 \\ 1.000 & 1.000 & -0.5000+0.8660 i & -0.5000-0.8660 i & -0.5000-0.8660 i & -0.5000+0.8660 i & 0 & 0 \\ 1.000 & -1.000 & -0.5000+0.8660 i & -0.5000-0.8660 i & 0.5000+0.8660 i & 0.5000-0.8660 i & 0 & 0 \\ 1.000 & -1.000 & -0.5000-0.8660 i & -0.5000+0.8660 i & 0.5000-0.8660 i & 0.5000+0.8660 i & 0 & 0 \\ \hline \end{array}

Representations of SL2(Z)SL_2(\mathbb{Z})

This fusion ring does not provide any representations of SL2(Z).SL_2(\mathbb{Z}).

Adjoint Subring

The adjoint subring is the ring itself.

The upper central series is trivial.

Universal grading

This fusion ring allows only the trivial grading.

Categorifications

Data

Download links for numeric data: